Aid agencies and NGOs have said Europe’s “unconscionable” response to the refugee crisis is courting humanitarian disaster, as Brussels scrambled to prepare emergency summits and desperate scenes unfolded across the continent, from Greece’s border with Macedonia to a makeshift camp outside Calais.

With the EU entering what many see as a make-or-break phase in tackling the crisis, the bloc’s most senior leaders called for urgent action to support Greece as at least 8,500 refugees and migrants remained trapped without permanent shelter on the country’s closed northern border with Macedonia.

Frontex, the EU’s border control agency, said 30 times as many migrants entered Europe in January and February as in the same two months of last year, and the UN’s refugee agency announced that 131,724 people had crossed the Mediterranean – the vast majority of them reaching Greece – so far in 2016, almost as many as made the journey in the first six months of 2015.

The UNHCR said the continent stood “on the cusp of a largely self-induced humanitarian crisis”, with governments “not working together despite agreements … and country after country imposing new border restrictions”.

In a scathing statement, Human Rights Watch condemned the EU’s “utter failure to respond collectively and compassionately to refugee flows”.

Germany’s chancellor, Angela Merkel, said Europe must deal decisively with “the difficult situation” in Greece, while the European council president, Donald Tusk, demanded support for Athens. Readiness to stand by Greece was “a test of our Europeanness”, Tusk said.

“The number of migrants arriving in Greece is on the rise not because they want to make Greece their home, but because they are hoping they will move on from Greece to other European countries,” he said as he embarked on a round of shuttle diplomacy before an emergency meeting with Turkey on Monday and a summit of EU leaders on 18 March.

Facing an average of 2,000-3,000 new refugee arrivals from Turkey each day, Greece said on Tuesday that the influx threatened to overwhelm its already overstretched resources and asked the EU for €480m (£375m) in emergency funds to help it shelter up to 100,000 refugees if needed.

“We cannot bear the strain of all the refugees coming here,” said Olga Gerovassili, a spokeswoman for the Greek government. “These are temporary measures, there needs to be a permanent solution on where refugees will be relocated.”

A hut burns as part of the ‘Jungle’ camp in Calais is cleared. Photograph: Carl Court/Getty Images

In France, meanwhile, migrants and refugees at the makeshift “Jungle” camp in Calais staged a sit-in protest and set light to shelters on Tuesday in an attempt to disrupt continuing demolition work at the site. Riot police fired more teargas overnight after an operation to dismantle hundreds of temporary shelters led to clashes on Monday.

In a fresh blow to free movement in the EU, Belgium announced it had turned back more than 600 migrants at the French border since the government had reinstated border controls owing to fears about the destruction of parts of the Calais camp, which houses up to 3,500 refugees and migrants eager to reach Britain and mostly unwilling to move into alternative accommodation elsewhere in France.

Greece was now home to at least 24,000 refugees and migrants needing accommodation, nearly one-third of them blocked on the border with Macedonia, said UNHCR spokesman Adrian Edwards. The Idomeni crossing remained closed on Tuesday after hundreds of migrants tried to force their way through on Monday, prompting police to fire stun grenades and teargas.

Charities at the border estimated that at least 2,000 more people had arrived in Idomeni on Tuesday. “They are coming by foot and as far as they can by bus and by taxi,” said Caroline Haga, of the International Red Cross Federation. “They are determined to leave. They absolutely do not want to stay in Greece.”

Syrians stranded at the frontier for the past week painted a picture of desperation. Amer Alabboush, 24, a chemical engineer from Homs, who reached Greece via Lesbos with three of his friends last week, said conditions at impromptu tent cities now spread out around the area had deteriorated dramatically.

“For the past week I have been sleeping outdoors because there are no tents available and I can tell you I am not the only one,” he said. “The situation is very bad, very difficult and it is getting worse. Me and my friends have been here for exactly seven days. Every day we have one sandwich. Nothing else. We are hungry.”

Refugees and migrants queue for food at a makeshift camp near Idomeni. Photograph: Louisa Gouliamaki/AFP/Getty Images

NGOs have been offering first aid from three medical tents, and long queues snaked through the camps as men, women and children lined up – often for hours – waiting for water and food. Sanitation amounted to a couple of showers and cabin toilets.

“If this continues, if the numbers get any bigger, it will be unbearable,” Haga said. “The queues are so long that people are passing out. Teams have treated over 100 cases today of refugees collapsing as they stand in line. We’ve just treated a woman who passed out waiting for water and food.”

Edwards said Athens “cannot manage this situation alone”. He said it was essential that EU member states honour pledges made last year to relocate more than 66,000 refugees from Greece over the next two years, noting that only 325 relocations had occurred so far.

The UNHCR said it “urges Greece and the states along the Balkans route to act quickly to avert a disaster and to approach this emergency in a spirit of solidarity and sharing of responsibility”.

In February Austria, Slovenia, Croatia, Serbia and Macedonia decided to impose tight border restrictions including daily caps on the number of asylum seekers and migrants allowed to enter their territory.

Macedonia’s president, Gjorge Ivanov, said in an interview with the German news site Spiegel Online that the entire Balkan refugee route would probably close once Austria reached its limit of 37,500 migrant entries this year. “In times of crisis, every government must find its own solution,” Ivanov said. “We can’t wait until Brussels makes a decision.”

Human Rights Watch said the chaos in northern Greece was a direct result of the caps, noting that Afghans in particular – the second-largest group making the hazardous journey to Greece – were now effectively stuck there while only a relatively small number of Syrian and Iraqi refugees each day were being allowed to pass into Macedonia and onwards into Serbia and Croatia.

“Trapping asylum seekers in Greece is an unconscionable and short-sighted non-solution,” said Eva Cossé, HRW’s Greece specialist. She said it was “ludicrous” that while the EU had agreed a relocation plan, “some countries’ actions risk turning Greece into a giant refugee camp”.

Merkel reiterated after meeting Croatia’s prime minister on Tuesday that the EU had to establish a mechanism to distribute refugees arriving in Greece to other European countries. “There is not a right for a refugee to say: ‘I want to get asylum in a particular country in the European Union,’” the chancellor said. Nearly 1.1 million people registered as asylum seekers in Germany last year.

Tusk, the European council president, set off on a three-day tour of six countries along the western Balkan route. Following a meeting on Wednesday with the Austrian chancellor, Werner Faymann, he will discuss the migration crisis with the leaders of Slovenia, Croatia, Macedonia, Greece and Turkey.

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Some of the most crucial meetings will be in Ankara, with Turkey’s prime minister, Ahmet Davutoğlu, and the president Recep Tayyip Erdoğan, before an EU-Turkey summit on Monday.

EU leaders have criticised Turkey, which is hosting 2.5 million refugees, for not doing enough to stem the flow of people heading to Greece, despite a €3bn deal signed last year aimed at preventing migrants from setting sail for Europe. In remarks aimed at Ankara, Tusk said he expected “more intensive engagement from our partners”.

EU leaders will hold a potentially crucial summit in mid-March in an attempt to forge new rules for managing asylum claims following the breakdown of the current system. They will discuss a contentious plan to scrap the Dublin rules, under which asylum seekers must register in the first country they arrive in.